f x = - 1 + k x + k neither touches nor intercepts the curve f x = ln ⁡ x , then minimum value of k ∈ is

Step-by-step solution

f x = - 1 + k x + k f x represents straight line with slope k and y -intercept k - 1 . f x + 1 = k x + 1 So for all values of k , it always passes through - 1 , - 1 For minimum slope of line, it will be touching the curve y = ln x When two curves touches, then at the common point their slopes are equal. f ' x = k , d y d x = 1 x k = 1 x   ⇒   x = 1 k ,   y = ln 1 k common point ⇒   1 k ,   - ln k Slope of line can be written from two points on the line. - ln k - - 1 1 k - -